It seems that upper and lower bounds in (REF ) reduce the difficulty for studying a random stepsize.
However, as revealed by the series of works in [1]}, [2]}, [3]}, [4]}, the analysis involving a random stepsize is intrinsically different from the one for non-adaptive schemes; additional terms arise due to the adaptivity; and adaptive schemes have promising practical benefits.
These differences inspire us to enable a random stepsize in the framework (under the restriction (REF )).
When \(\chi _t = 0\) , we arrive at a non-adaptive scheme.
